#8324416

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#42232	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3899 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,7,10)(3,6,9,15)(4,5,8,14)(11,12,13,16)(17,18)(19,28)(20,22)(21,31)(23,27)(24,30)(25,29)(26,32)(34,39)(36,42)(40,46)(44,48);(1,3)(2,6)(4,11)(5,12)(7,9)(8,13)(10,15)(14,16)(17,19,23,30)(18,24,27,28)(20,21,29,32)(22,26,25,31)(35,43)(40,46)(44,48)(45,47);(3,9)(6,15)(11,13)(12,16)(17,20,23,29)(18,25,27,22)(19,21,30,32)(24,31,28,26)(33,34)(35,40)(36,41)(37,42)(38,39)(43,46)(44,47)(45,48);(1,4)(2,5)(3,11)(6,12)(7,8)(9,13)(10,14)(15,16)(17,19)(18,24)(20,21)(22,26)(23,30)(25,31)(27,28)(29,32)(33,35)(34,40)(36,44)(37,45)(38,43)(39,46)(41,47)(42,48);(1,5)(2,8)(3,12)(4,10)(6,13)(7,14)(9,16)(11,15)(17,21)(18,26)(19,29)(20,30)(22,28)(23,32)(24,25)(27,31)(33,36,38,42)(34,41,39,37)(35,44,43,48)(40,47,46,45);(1,6,7,15)(2,9,10,3)(4,12,8,16)(5,13,14,11)(17,22,23,25)(18,20,27,29)(19,26,30,31)(21,28,32,24)(33,37,38,41)(34,42,39,36)(35,45,43,47)(40,48,46,44)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8231612);